Ms Brennan argued the broadcast was "extremely distasteful" because of its sexual connotations and meaning.
But the BSA disagreed, taking the same point of view as MediaWorks, which owns the station, that the content was not explicitly sexual.
"We agree with MediaWorks that the broadcast did not contain any explicit sexual content and that the innuendo would have gone over the heads of most children.
"While recognising the station targets 15 to 39-year olds rather than children, we do not think the segment would have caused alarm or distress to any children who happened to be listening."
MediaWorks said The Edge was not targeted at children and was well-known for content that could sometimes be risque.
Hosts Jay-Jay Feeney, Mike Puru and Dom Harvey were "renowned for their wit and quirky senses of humour; their light-hearted banter and for regularly engineering mischievous but innocuous pranks and jokes", MediaWorks said.
It's not the first time the trio have been under fire over their on-air antics.
Earlier this year, Harvey was criticised for tweeting an inappropriate screenshot of Bachelorette Chrystal Chenery, who competed alongside his wife on Dancing With The Stars.
In June, Gaynz.com reported that The Edge was planning a "Trans for the day" segment, which would see Dom Harvey made over as a woman.
Despite being urged by transgender advocates to call off the stunt, The Edge changed the wording slightly and it went ahead.
In 2013, Dom Harvey was slammed for connecting child rape with a contestant on the X-Factor, writing: "Poor Gracie! First molested in her own bed by uncle bully. And now kicked out of #xfactornz" in an apparent reference to the film Once Were Warriors.